A DIMENSION DROP PHENOMENON OF FRACTAL CUBES LIANG-YI HUANG AND HUI RAO∗ Abstract. Let E be a metric space. We introduce a notion of connectedness index of E, which is the Hausdorff dimension of the union of non-trivial connected components of E. We show that the connectedness index of a fractal cube E is strictly less than the Hausdorff dimension of E provided that E possesses a trivial connected component. Hence the connectedness index is a new Lipschitz invariant. Moreover, we investigate the relation between the connectedness index and topological Hausdorff dimension. 1. Introduction N d An iterated function system (IFS) is a family of contractions f'jgj=1 on R , and the attractor of the IFS is the unique nonempty compact set K satisfying SN K = j=1 'j(K), and it is called a self-similar set [5]. Let n ≥ 2 and let D = d fd1; ··· ; dN g ⊂ f0; 1; : : : ; n−1g , which we call a digit set. Denote by #D := N the 1 N cardinality of D. Then n and D determine an IFS f'j(z) = n (z + dj)gj=1, whose attractor E = E(n; D) satisfies the set equation 1 (1.1) E = (E + D): n We call E a fractal cube [12], especially, when d = 2, we call E a fractal square [6]. There are some works on topological and metric properties of fractal cubes. Why- burn [11] studied the homeomorphism classification, Bonk and Merenkov [2] studied the quasi-symmetric classification. Lau, Luo and Rao [6] studied when a fractal square is totally disconnected. Xi and Xiong [12] gave a complete classification of Lipschitz equivalence of fractal cubes which are totally disconnected. Recently, the arXiv:2010.12890v1 [math.GN] 24 Oct 2020 studies of [10,13] focus on the the Lipschitz equivalence of fractal squares which are not totally disconnected. Topological Hausdorff dimension is a new fractal dimension introduced by Buc- zolich and Elekes [1]. It is shown in [1] that for any set K we always have dimtH K ≤ dimH K, where dimtH and dimH denote the topological Hausdorff dimension and Hausdorff dimension respectively. Ma and Zhang [8] calculated topological Haus- dorff dimensions of a class of fractal squares. Date: October 12, 2020. The work is supported by NSFS Nos. 11971195 and 11601172. 2000 Mathematics Subject Classification: 28A80,26A16 Key words and phrases: fractal cube, connected component, topological Hausdorff dimension . * The correspondence author. 1 Let K be a metric space. A point x 2 K is called a trivial point of K if fxg is a connected component of K. Let Λ(K) be the collection of trivial points in K. Denote (1.2) Ic(K) := dimH K n Λ(K); and we call it the connectedness index of K. It is obvious that Ic(K) ≤ dimH K. Clearly, the connectedness index is a Lipschitz invariant. The main results of the present paper are as follows. Theorem 1.1. Let E = E(n; D) be a d-dimensional fractal cube. If E has a trivial point, then Ic(E) < dimH E. However, Theorem 1.1 is not valid for general self-similar sets, even if the self- similar sets satisfy the open set condition. 1 S 1 1 Q 1 Example 1.1. Let Q = f0g [ [ 22k+1 ; 22k ] . Observe that Q = 4 [ [ 2 ; 1] and k=0 Q 2 [ Q = [0; 1]. Then Q is a self-similar set satisfying the equation Q Q 1 Q 1 Q = [ + [ + : 4 4 2 2 2 The set Q has only one trivial point, that is 0. Therefore, Ic(Q) = dimH Q = 1. Figure1 illustrates Q0, a two dimensional generalization of Q. Similarly, Q0 is a self-similar set, and the unique trivial point of Q0 is 0. Figure 1. The self-similar set Q0. Using Theorem 3.7 of [1] we show the following. Theorem 1.2. For a non-empty σ-compact metric space K, we have dimtH K ≤ Ic(K). Zhang [14] asked when dimtH E = dimH E, where E is a fractal square. According to [14], a digit set D is called a Latin digit set, if every row and every column has the same number of elements (see Figure2). For a fractal square E = E(n; D), Zhang showed that if dimtH E = dimH E, then either E = [0; 1] × C, or E = C × [0; 1] for some C ⊂ [0; 1], or D is a Latin digit set. As a corollary of Theorem 1.1 and Theorem 1.2, we obtain a new necessary con- dition for dimH E = dimtH E. 2 (a) The digit set of L. (b) The Latin fractal square L. Figure 2. It is shown in [14] that log 12= log 6 = dimtH L < dimH L = log 24= log 6. While by Theorem 1.1 and Theorem 1.2, we directly have dimtH L < dimH L. Corollary 1.1. Let E be a d-dimensional fractal cube. If dimH E = dimtH E, then E has no trivial point. Remark 1.1. Another application of Theorem 1.1 is on the gap sequences of fractal cubes, a Lipschitz equivalent invariant introduced by Rao, Ruan and Yang [9]. For a fractal cube K, let fgm(K)gm≥1 be the gap sequence. Using Theorem 3.1 of the present paper, it is proved in [4] that if K has trivial point, then fgm(K)gm≥1 is −1/γ equivalent to fm gm≥1, where γ = dimH K. Finally, we calculate the connectedness indexes of two fractal squares in Figure3, and illustrate the application to Lipschitz classification. Example 1.2. Let K and K0 be two fractal squares indicated by Figure3. It is 0 log 14 seen that dimH K = dimH K = log 5 . By Theorem 1.3 of [8], one can obtain that 0 log 2 dimtH K = dimtH K = 1 + log 5 . We will show in section 5 that p log(8 + 132=2) log 13 I (K) = and I (K0) = : c log 5 c log 5 So K and K0 are not Lipschitz equivalent. (a) The digit set of K. (b) The digit set of K0. Figure 3. 3 This article is organized as follows. In section 2, we recall some basic facts of r-face of the polytope [0; 1]d. In section 3, we prove Theorem 1.1. In section 4, we prove Theorem 1.2. In section 5, we give the details of Example 1.2. 2. Preliminaries on r-faces of [0; 1]d We recall some notions about convex polytopes, see [15]. Let C ⊂ Rd be a convex polytope, let F be a convex subset of C. The affine hull of F , denoted by aff(F ), is the smallest affine subspace containing F . We say F is a face of C, if any closed line segment in C with a relative interior in F has both endpoints in F . The dimension of an affine subspace is defined to be the dimension of the corre- sponding linear vector space. The dimension of a face F , denoted by dim F , is the dimension of its affine hull. Moreover, F is called an r-face of C, if F is a face of C with dimension r. We take it by convention that C is a d-face of itself if dim C = d. For z 2 C, a face F of C is called the containing face of z if z is a relative interior point of F . d Let e1;:::; ed be the canonical basis of R . The following facts about the r-faces of [0; 1]d are obvious, see Chapter 2 of [15]. Lemma 2.1. (i) Let A [ B = f1; : : : ; dg be a partition with #A = r. Then the set ( ) X (2.1) F = cjej; cj 2 [0; 1] + b j2A is an r-face of [0; 1]d if and only if b 2 T , where ( ) X (2.2) T := "jej; "j 2 f0; 1g ; j2B (ii) For any r-face F of [0; 1]d, there exists a partition A [ B = f1; : : : ; dg with #A = r such that F can be written as (2.1). P We will call F0 = f cjej; cj 2 [0; 1]g a basic r-face related to the partition j2A A [ B. We give a partition B = B0 [ B1 according to b by setting B = fj 2 B; the j-th coordinate of b is 0g; (2.3) 0 B1 = fj 2 B; the j-th coordinate of b is 1g: P P d Let x = αjej + βiei 2 [0; 1] , we define two projection maps as follows: j2A i2B X X (2.4) πA(x) = αjej; πB(x) = βiei: j2A i2B If F is an r-face of [0; 1]d, we denote by F˚ the relative interior of F . Lemma 2.2 (Chapter 2 of [15]). Let C ⊂ Rd be a polytope. (i) If G and F are faces of C and F ⊂ G, then F is a face of G. (ii) If G is a face of C, then any face of G is also a face of C. 4 The following lemma will be needed in section 3. d d Lemma 2.3. Let F = F0 +b be an r-face of [0; 1] given by (2.1). Let u 2 Z . Then F˚ \ (u + [0; 1]d) 6= ; if and only if u = b − b0 for some b0 2 T , where T is defined in (2.2). 0 0 0 d Proof. \(": Suppose b 2 T , then F − (b − b ) = F0 + b , and it is an r-face of [0; 1] by Lemma 2.1 (i).